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Conformal Oscillator Representations of Orthogonal Lie Algebras

The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n+2). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a, we prove that the space forms an irreducible o(n+2)-module for any constant c if the vector a is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2) on the polynomial algebra C in n variables. Moreover, we prove that the algebra C forms an infinite-dimensional irreducible weight o(n+2)-module with finite-dimensional weight subspaces if the constant c is not a half integer.